a) $f_Y (y | X = x) =$ ?
$\,\,\,\,\,\,\,E (y | X = x) =$ ?
b) $f(x,y) =$ ?
c) $f_Y (y) =$ ?
d) $f_X (x | Y = y) =$ ?
e) $E(X | Y = y) =$ ?
My work:
a) $f_Y(y|x) = \frac{1}{(x-0)} = \frac{1}{x}\\E(y|x) = \int_0^x y f(y|x)dy = \int_0^x\frac{y}{x} = \frac{x}{2}$
b) $f(x,y) = f(y|x)f(x) = \frac{1}{x}( xe^{-x}) = e^{-x}$
c) $f_Y(y) = \int_0^\infty f(x,y)dx = \int_0^\infty e^{-x} dx = 1$
d) $f_X(x|y)= \frac{f(x,y)}{f(y)} = e^{-x}$
e) $E(x|y) = \int_0^\infty x f(x|y) dx = \int_0^\infty x e^{-x} dx$ Which is the mean with exponential distribution so, $E(x|y) = \frac{1}{\lambda} = \frac{1}{1} = 1$
Does my work seem to be correct?
a) $f_Y(y|X=x)=\frac1x\mathbf 1_{0\lt y\lt x}$ , $E(Y|X=x)=\frac12x$
b) $f(x,y)=\mathrm e^{-x}\mathbf 1_{0\lt y\lt x}$
c) $f_Y(y)=\mathrm e^{-y}\mathbf 1_{y\gt0}$
d) $f_X(x|Y=y)=\mathrm e^{-(x-y)}\mathbf 1_{x\gt y}$
e) $E(X|Y=y)=y+1$